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Global solvability and hypoellipticity on the torus for a class of differential operators with variable coefficients

Technical Report:

Abstract

The present paper studies global hypoellipticity and global solvability on the two dimensional torus T{sup 2} for a class of second order differential operators with variable coefficients of the type P=(D{sub x}-ia(x)D{sub y}) (D{sub x}-ib(x)D{sub y})+(a`(x)-b`(x))D{sub y}+c(x). Necessary and/or sufficient conditions for global solvability and global hypoellipticity are proposed. In particular if Rea(x) {identical_to} 0 and/or Reb(x) {identical_to} 0 Siegel type conditions on the diophantine approximation of the averages {integral}{sub 0}{sup 2{pi}}Rea(x)dx or/and {integral}{sub 0}{sup 2{pi}}Reb(x)dx occur. We also indicate some results for more general class of operators and for the n-dimensional torus T{sup n}, n > 2. (author). 18 refs.
Authors:
Gramchev, T; [1]  Popivanov, P; [2]  Yoshino, M [3] 
  1. International Centre for Theoretical Physics, Trieste (Italy)
  2. Bylgarska Akademiya na Naukite, Sofia (Bulgaria). Matematischeski Inst.
  3. Chuo Univ., Tokyo (Japan)
Publication Date:
Jul 01, 1991
Product Type:
Technical Report
Report Number:
IC-91/190
Reference Number:
SCA: 661300; PA: AIX-23:015348; SN: 92000647065
Resource Relation:
Other Information: PBD: Jul 1991
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DIFFERENTIAL EQUATIONS; ANALYTICAL SOLUTION; DIFFERENTIAL GEOMETRY; MATHEMATICAL OPERATORS; 661300; OTHER ASPECTS OF PHYSICAL SCIENCE
OSTI ID:
10113318
Research Organizations:
International Centre for Theoretical Physics (ICTP), Trieste (Italy)
Country of Origin:
IAEA
Language:
English
Other Identifying Numbers:
Other: ON: DE92615228; TRN: XA9130243015348
Availability:
OSTI; NTIS (US Sales Only); INIS
Submitting Site:
INIS
Size:
13 p.
Announcement Date:
Jun 30, 2005

Technical Report:

Citation Formats

Gramchev, T, Popivanov, P, and Yoshino, M. Global solvability and hypoellipticity on the torus for a class of differential operators with variable coefficients. IAEA: N. p., 1991. Web.
Gramchev, T, Popivanov, P, & Yoshino, M. Global solvability and hypoellipticity on the torus for a class of differential operators with variable coefficients. IAEA.
Gramchev, T, Popivanov, P, and Yoshino, M. 1991. "Global solvability and hypoellipticity on the torus for a class of differential operators with variable coefficients." IAEA.
@misc{etde_10113318,
title = {Global solvability and hypoellipticity on the torus for a class of differential operators with variable coefficients}
author = {Gramchev, T, Popivanov, P, and Yoshino, M}
abstractNote = {The present paper studies global hypoellipticity and global solvability on the two dimensional torus T{sup 2} for a class of second order differential operators with variable coefficients of the type P=(D{sub x}-ia(x)D{sub y}) (D{sub x}-ib(x)D{sub y})+(a`(x)-b`(x))D{sub y}+c(x). Necessary and/or sufficient conditions for global solvability and global hypoellipticity are proposed. In particular if Rea(x) {identical_to} 0 and/or Reb(x) {identical_to} 0 Siegel type conditions on the diophantine approximation of the averages {integral}{sub 0}{sup 2{pi}}Rea(x)dx or/and {integral}{sub 0}{sup 2{pi}}Reb(x)dx occur. We also indicate some results for more general class of operators and for the n-dimensional torus T{sup n}, n > 2. (author). 18 refs.}
place = {IAEA}
year = {1991}
month = {Jul}
}